3.860 \(\int \frac{1}{x \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=63 \[ -\frac{b \log \left (a+b x^n\right )}{a n (b c-a d)}+\frac{d \log \left (c+d x^n\right )}{c n (b c-a d)}+\frac{\log (x)}{a c} \]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^n])/(a*(b*c - a*d)*n) + (d*Log[c + d*x^n])/(c*(b*c
- a*d)*n)

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Rubi [A]  time = 0.18569, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b \log \left (a+b x^n\right )}{a n (b c-a d)}+\frac{d \log \left (c+d x^n\right )}{c n (b c-a d)}+\frac{\log (x)}{a c} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^n)*(c + d*x^n)),x]

[Out]

Log[x]/(a*c) - (b*Log[a + b*x^n])/(a*(b*c - a*d)*n) + (d*Log[c + d*x^n])/(c*(b*c
- a*d)*n)

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Rubi in Sympy [A]  time = 27.2799, size = 49, normalized size = 0.78 \[ - \frac{d \log{\left (c + d x^{n} \right )}}{c n \left (a d - b c\right )} + \frac{b \log{\left (a + b x^{n} \right )}}{a n \left (a d - b c\right )} + \frac{\log{\left (x^{n} \right )}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*x**n)/(c+d*x**n),x)

[Out]

-d*log(c + d*x**n)/(c*n*(a*d - b*c)) + b*log(a + b*x**n)/(a*n*(a*d - b*c)) + log
(x**n)/(a*c*n)

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Mathematica [A]  time = 0.0858214, size = 56, normalized size = 0.89 \[ \frac{-b c \log \left (a+b x^n\right )+a d \log \left (c+d x^n\right )-a d n \log (x)+b c n \log (x)}{a b c^2 n-a^2 c d n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^n)*(c + d*x^n)),x]

[Out]

(b*c*n*Log[x] - a*d*n*Log[x] - b*c*Log[a + b*x^n] + a*d*Log[c + d*x^n])/(a*b*c^2
*n - a^2*c*d*n)

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Maple [A]  time = 0.015, size = 69, normalized size = 1.1 \[{\frac{\ln \left ({x}^{n} \right ) }{anc}}+{\frac{b\ln \left ( a+b{x}^{n} \right ) }{an \left ( ad-bc \right ) }}-{\frac{d\ln \left ( c+d{x}^{n} \right ) }{nc \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n)/(c+d*x^n),x)

[Out]

1/n/a/c*ln(x^n)+1/n*b/a/(a*d-b*c)*ln(a+b*x^n)-1/n*d/c/(a*d-b*c)*ln(c+d*x^n)

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Maxima [A]  time = 1.38794, size = 93, normalized size = 1.48 \[ -\frac{b \log \left (\frac{b x^{n} + a}{b}\right )}{a b c n - a^{2} d n} + \frac{d \log \left (\frac{d x^{n} + c}{d}\right )}{b c^{2} n - a c d n} + \frac{\log \left (x\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)*x),x, algorithm="maxima")

[Out]

-b*log((b*x^n + a)/b)/(a*b*c*n - a^2*d*n) + d*log((d*x^n + c)/d)/(b*c^2*n - a*c*
d*n) + log(x)/(a*c)

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Fricas [A]  time = 0.241621, size = 78, normalized size = 1.24 \[ -\frac{b c \log \left (b x^{n} + a\right ) - a d \log \left (d x^{n} + c\right ) -{\left (b c - a d\right )} n \log \left (x\right )}{{\left (a b c^{2} - a^{2} c d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)*x),x, algorithm="fricas")

[Out]

-(b*c*log(b*x^n + a) - a*d*log(d*x^n + c) - (b*c - a*d)*n*log(x))/((a*b*c^2 - a^
2*c*d)*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*x**n)/(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)*(d*x^n + c)*x),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)*x), x)